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This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations. The book begins with a conceptual formulation of the Gross-Zagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of L-series by means of Weil representations. The Gross-Zagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the Gross-Zagier formula is reduced to local formulas. The Gross-Zagier Formula on Shimura Curves will be of great use to students wishing to enter this area and to those already working in it.

Shimura varieties. --- Arithmetical algebraic geometry. --- Automorphic forms. --- Quaternions. --- Algebra, Universal --- Algebraic fields --- Curves --- Surfaces --- Numbers, Complex --- Vector analysis --- Automorphic functions --- Forms (Mathematics) --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Number theory --- Varieties, Shimura --- Arithmetical algebraic geometry --- Arakelov theory. --- Benedict Gross. --- Don Zagier. --- EichlerГhimura theory. --- Eisenstein series. --- GrossКagier formula. --- Heegner point. --- Hodge bundle. --- Hodge index theorem. --- L-series. --- MordellЗeil group. --- NeronДate height. --- RankinГelberg L-function. --- Schwartz function. --- Shimizu lifting. --- Shimura curve. --- Shimura curves. --- SiegelЗeil formula. --- Waldspurger formula. --- Weil representation. --- abelian varieties. --- analytic kernel function. --- analytic kernel. --- degenerate Schwartz function. --- discrete series. --- generating series. --- geometric kernel. --- height series. --- holomorphic kernel function. --- holomorphic projection. --- incoherent Eisenstein series. --- incoherent automorphic representation. --- incoherent quaternion algebra. --- kernel function. --- kernel identity. --- local height. --- modular curve. --- modularity. --- multiplicity function. --- non-archimedean local field. --- non-degenerate quadratic space. --- ordinary component. --- orthogonal space. --- projector. --- pull-back formula. --- ramified quadratic extension. --- supersingular component. --- superspecial component. --- theta function. --- theta liftings. --- theta series. --- trace identity. --- un-normalized kernel function. --- unramified quadratic extension.